Homogenization of supremal functionals in the vectorial case (via $L^p$-approximation)

We propose a homogenized supremal functional rigorously derived via $L^p$-approximation by functionals of the type $f(\frac{x}{\epsilon },Du)$, when $\mathrm{\Omega }$ is a bounded open set of ${\mathbb{R}}^n$ and $u\in W^{1,\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^d)$. The homogenized functional is also deduced directly in the case where the sublevel sets of $f(x,\cdot )$ satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals.